Linear independence of linear forms in polylogarithms

Opaque sales are a selling mechanism in which a seller offers multiple products but the buyer does not know which specific product they will receive until after the purchase. This mechanism is widely used in the tourism industry, such as staycation packages with undisclosed hotel options or tours to different destinations, and in e-commerce through mystery boxes. In many such settings, buyers operate under a unit demand constraint, meaning they derive benefit from only one item despite multiple options being available. In this paper, we formulate and solve a multi-item mechanism design problem for opaque sales. Our goal is to identify a distribution-free mechanism that maximizes the competitive ratio, defined as the ratio between the revenue generated by the mechanism and the maximum revenue attainable with full knowledge of the buyer's valuations. Despite the problem's infinite dimensionality, we show that the optimal mechanism admits a semi-analytical form, characterized by the solution of an auxiliary convex optimization problem whose size scales linearly with the number of items. Furthermore, we demonstrate that this mechanism can be equivalently implemented through a menu of infinite level-access lotteries, where the buyer pays an amount to access a randomly assigned subset of items, with higher payments granting access to a larger selection. We then analyze the impact of menu size limitations on the competitive ratio and conclude with a prototypical example illustrating our approach.

Using the theory of Dirichlet forms on topological vector spaces we construct solutions to stochastic differential equations in infinite dimensions of the type $$dX_t = dW_t + \beta (X_t )dt$$ for possibly very singular drifts β. Here (X t ) t ≧0 takes values in some topological vector spaceE and (W t ) t ≧0 is anE-valued Brownian motion. We give applications in detail to (infinite volume) quantum fields where β is e.g. a renormalized power of a Schwartz distribution. In addition, we present a new approach to the case of linear β which is based on our general results and second quantization. We also prove new results on general diffusion Dirichlet forms in infinite dimensions, in particular that the Fukushima decomposition holds in this case.

On the learnability of vector spaces

The convergence properties for the solution of the discrete time Riccati matrix equation are extended to Riccati operator equations such as arise in a gyroscope noise filtering problem. Stabilizability and detectability are shown to be necessary and sufficient conditions for the existence of a positive semidefinite solution to the algebraic Riccati equation which has the following properties (i) it is the unique positive semidefinite solution to the algebraic Riccati equation, (ii) it is converged to geometrically in the operator norm by the solution to the discrete Riccati equation from any positive semidefinite initial condition, (iii) the associated closed loop system converges uniformly geometrically to zero and solves the regulator problem, and (iv) the steady state Kalman–Bucy filter associated with the solution to the algebraic Riccati equation is uniformly asymptotically stable in the large. These stability results are then generalized to time-varying problems; also it is shown that even in infinite dimensions, controllability implies stabilizability.

For every complex number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar x double-vertical-bar Subscript double-struck upper Z Baseline colon equals min left-brace right-brace colon vertical-bar vertical-bar minus minus xm colon separator element-of element-of mZ"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>x</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>:=</mml:mo> <mml:mo movablelimits="true" form="prefix">min</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mo> − </mml:mo> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>m</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Vert x\Vert _{\mathbb {Z}}:=\min \{|x-m|:\ m\in \mathbb {Z}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a number field, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">k\in \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha 1 comma ellipsis comma alpha Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha _1,\ldots ,\alpha _k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta element-of left-parenthesis 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> θ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\theta \in (0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that there are infinitely many tuples <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n comma q 1 comma ellipsis comma q Subscript k Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(n,q_1,\ldots ,q_k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar q 1 alpha 1 Superscript n Baseline plus midline-horizontal-ellipsis plus q Subscript k Baseline alpha Subscript k Superscript n Baseline double-vertical-bar Subscript double-struck upper Z Baseline greater-than theta Superscript n"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msubsup> <mml:mi> α </mml:mi> <mml:mn>1</mml:mn> <mml:mi>n</mml:mi> </mml:msubsup>

If $\alpha$ is a non-zero algebraic number, we let $m(\alpha)$ denote the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the $t$-metric Mahler measure, denoted $m_t(\alpha)$. For fixed $\alpha\in \bar{\mathbb Q}$, the map $t\mapsto m_t(\alpha)$ is continuous, and moreover, is infinitely differentiable at all but finitely many points, called {\it exceptional points} for $\alpha$. It remains open to determine whether there is a sequence of elements $\alpha_n\in \bar{\mathbb Q}$ such that the number of exceptional points for $\alpha_n$ tends to $\infty$ as $n\to \infty$. We utilize a connection with the Fibonacci sequence to formulate a conjecture on the $t$-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results.

An asymptotic, large time solution of the convection Stefan problem with surface radiation

For any compact oriented surface $\Sigma$ we consider the group of diffeomorphisms of $\Sigma$ which preserve a given area form. In this paper we show that the vector space of homogeneous quasi-morphisms on this group has infinite dimension. This result is proved by constructing explicitly and for each surface an infinite family of independent homogeneous quasi-morphisms. These constructions use simple arguments related to linking properties of the orbits of the diffeomorphisms.

Chapter 17 - Metric Geometry

On the Cauchy problem for non-local Ornstein–Uhlenbeck operators

Geophysical Inverse Theory and Regularization Problems

The method of Levi Civita is applied to an isolated fully cavitating body at zero cavitation number and adapted to the solution of the inverse problem in which one prescribes the pressure distribution on the wetted surface and then calculates the shape. The novel feature of this work is the finding that the exact theory admits the existence of a “point drag” function or eigensolution. While this fact is of no particular importance in the classical direct problem, we already know from the linearized theory that the eigensolution plays an important role. In the present discussion, the basic properties of the exact “point-drag” solution are explored under the simplest of conditions. In this way, complications which arise from non-zero cavitation numbers, free surface effects, or cascade interactions are avoided. The effects of this simple eigensolution on hydrodynamic forces and cavity shape are discussed. Finally, we give a tentative example of how this eigensolution might be used in the design process.

A large number of tasks in the economy are aimed at processing little formalized data. These can be tasks of analysis, planning, forecasting in the economy. The solution of these problems is reduced to the formalization of the data flow into a functional dependence. The definition of functional dependence is reduced to the approximation of data by a mathematical function. It is convenient to represent poorly formalized data as fuzzy numbers and reduce the solution of the approximation problem to the calculation of systems of fuzzy linear equations. This is achieved by developing the apparatus of software systems operating with calculations of fuzzy matrices. We have implemented computational approaches using the Zadeh principle and ordered fuzzy numbers.

Let G be a subgroup of the group GL ( V, F ) of automorphisms of a vector space V over a field F . By definition, the augmentation dimension of G , augdim F G for short, is the F -dimension of the vector subspace V (ω FG ), where ω FG is the augmentation ideal of the group ring FG . It is well known that G is finitary if and only if every finitely generated subgroup of G has finite augmentation dimension. In the antipodes of this concept, we define G to be antifinitary , if every proper subgroup of G of infinite augmentation dimension is finitely generated. In this paper we describe some generalized soluble antifinitary groups. 2000 Mathematics Subject Classification: 20F22; 20H20.

It is proved that if a Schur superalgebra is not semisimple, then it is neither cellular nor quasi-hereditary (Theorem 2), and it has infinite global dimension (Corollary 18). The algebra S(m|n, r) with m, n ⩾ 1 is semisimple if and only if p, the characteristic of the ground field, is zero or greater than r, or when m = n = 1 and p does not divide r. 2000 Mathematics Subject Classification 17A70 (primary), 20C30 (secondary).